Who This Page Is For
Linear algebra serves three very different audiences. The same course feels different depending on which one you are — and so does the tutoring.
Engineering & CS majors
You need fluency with matrices, vector spaces, and eigenvalues for downstream courses — controls, graphics, ML, signal processing. We focus on intuition + speed.
Math majors writing proofs
Your class is half computation, half proof, and the proofs feel disconnected from the calculations. We bridge the two — every proof starts from a picture and every picture has a proof.
Data science & ML learners
You want the linear algebra that actually shows up in ML: SVD, PCA, projections, gradient geometry, least squares. We skip the parts you'll never use and double down on the parts you'll use every week.
What We Cover
Standard linear algebra syllabus, plus the data science / ML connections most textbooks skip.
Vectors & Geometry
- Vectors, magnitude, direction
- Dot product, cross product
- Projections, orthogonality
- Linear combinations and span
- Parametric lines and planes
Matrices & Systems
- Row reduction (RREF)
- Solving Ax = b
- Matrix multiplication intuition
- Inverses, determinants
- LU, QR factorization
- Block matrices
Vector Spaces & Theory
- Linear independence, basis, dimension
- Rank, nullity, rank-nullity theorem
- Four fundamental subspaces
- Change of basis
- Coordinate systems
- Proof techniques and structure
Transformations & Eigenvalues
- Linear transformations
- Eigenvalues and eigenvectors
- Diagonalization
- Similarity, characteristic polynomial
- Markov chains, dynamics
Orthogonality & Projections
- Gram-Schmidt
- Orthogonal projections
- Least squares regression
- Symmetric matrices, spectral theorem
- Quadratic forms
Advanced & Applied
- Singular Value Decomposition (SVD)
- Principal Component Analysis (PCA)
- Pseudoinverse
- Numerical stability and conditioning
- Applications to ML, graphics, statistics
How I'd Walk You Through Finding Eigenvalues
Eigenvalues confuse students because the algebra is mechanical but the meaning is hidden. Here's how I'd connect both in a session.
- Start with the definition. An eigenvector v satisfies Av = λv. That means A acts on v just like multiplication by a scalar — it stretches but doesn't rotate it. λ is the stretch factor.
- Rearrange. Av = λv becomes (A − λI)v = 0. For a nonzero v to satisfy this, the matrix (A − λI) must be singular — meaning det(A − λI) = 0.
- Set up the characteristic polynomial. A − λI = [[4−λ, 1], [2, 3−λ]]. Take its determinant: (4−λ)(3−λ) − 2 = 0.
- Expand and solve. λ² − 7λ + 10 = 0, which factors as (λ − 2)(λ − 5) = 0. So λ = 2 and λ = 5.
- Sanity check. Trace = sum of eigenvalues: 4 + 3 = 2 + 5 ✓. Determinant = product: 4·3 − 1·2 = 10 = 2·5 ✓. Two free checks that catch arithmetic mistakes.
Where Linear Algebra Students Usually Get Stuck
"I can row-reduce, but I don't know what it means"
RREF is mechanical. We work on what RREF is doing geometrically and why pivots map onto dimension, rank, and linear independence.
"The proofs feel unrelated to the calculations"
They're not — proofs are just careful statements about what the calculations always do. We translate between the two until they feel like the same skill.
"I memorized eigenvalue steps but have no idea why"
We rebuild the intuition first: an eigenvector is a direction the matrix doesn't rotate. From there, every formula becomes a consequence, not a memorization.
"I need linear algebra for ML and my class isn't teaching that"
We cover SVD, projections, gradients, and PCA the way they're actually used in machine learning, not just as exam material.
Frequently Asked Questions
What linear algebra topics can you tutor?
Vectors, dot and cross products, matrices, row reduction, determinants, vector spaces, linear independence, basis, dimension, rank, null space, linear transformations, eigenvalues and eigenvectors, diagonalization, orthogonality, projections, least squares, QR factorization, SVD, and proof writing.
Can you help with proofs?
Yes. Proof writing is one of the trickiest parts of linear algebra and we spend dedicated time on it — recognizing definitions, structuring an argument, and writing it cleanly.
I need linear algebra for machine learning. Can you help with that angle?
Absolutely. We connect every concept — eigenvalues, projections, SVD, gradients — directly to where it shows up in ML, statistics, and data science, not just abstract problem sets.
What software or notation do you use?
We use whichever your course uses — standard mathematical notation, plus Python/NumPy or MATLAB when it helps you visualize. Sessions run on a shared online whiteboard.